To calculate the angle $\theta$ in the triangle, where:

• Opposite side to $\theta$ = $53.4\ \text{m}$,
• Adjacent side to $\theta$ = $79.6\ \text{m}$,

we can use the tangent function in trigonometry:

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}.$

Step-by-Step Solution:

1. Substitute the given values into the formula:

$\tan(\theta) = \frac{53.4}{79.6}.$

2. Calculate the ratio:

$\tan(\theta) = 0.6707.$

3. Use the inverse tangent function ($\tan^{-1}$) to find $\theta$:

$\theta = \tan^{-1}(0.6707).$

4. Calculate $\theta$ using a calculator:

$\theta = 33.8^\circ \ \text{(to 1 decimal place)}.$

Final Answer:

The size of angle $\theta$ is 33.8°.

Would you like a more detailed explanation or verification? Let me know!
Here are 5 related questions to explore further:

1. How do you calculate angles in right-angled triangles using sine and cosine?
2. What is the difference between $\sin$, $\cos$, and $\tan$ ratios in trigonometry?
3. How do you solve for unknown sides in a right triangle using trigonometric functions?
4. What is the Pythagorean theorem, and how can it help in right triangles?
5. How can trigonometry be applied to real-world problems like surveying or construction?

Tip: Always make sure your calculator is in degree mode when solving for angles!